1. Field of the Invention
The present invention relates to the cancellation of the effects caused by the wavelength dependence of the nonlinearity coefficient of highly nonlinear microstructured fibers in all-optical signal processing applications.
2. Description of the Related Arts
Nonlinear phenomena in optical waveguides (including optical fibers) enable all-optical signal processing, which can overcome many limitations given by conventional signal processing in the electrical domain. These nonlinear phenomena include effects such as self-phase modulation, cross-phase modulation, four wave mixing and Raman amplification.
One of the most promising nonlinear waveguides are small-core microstructured optical fibers, also called photonic crystal fibers (PCFs). The high index contrast between core and cladding in PCFs strongly confines the light in the core region. Thus, a small core and the use of a core material with a high nonlinear refractive index allow the realization of fibers with a nonlinearity coefficient γ well above that of conventional highly nonlinear fibers. Increasing the nonlinearity coefficient reduces the length required to achieve a certain magnitude of nonlinearity. This relaxes requirements regarding the chromatic dispersion characteristics, which limit the wavelength range of various all-optical signal processing applications such as cross-phase modulation (see non-patent document 1), four wave mixing (FWM) (see non-patent document 2) and parametric amplification. However, another limitation that so far has not been overlooked arises due to the wavelength dependence of the nonlinearity coefficient.
In the following, equations valid in the case of self-(SPM) and cross-phase modulation (XPM) will be given. The nonlinear phase shift experienced by light of the wavelength λi propagating through a fiber with the length L is given by (see non-patent document 3),φ(λi)=bγ(λi,λj)PjLeff(λj)  (1)where Pj is the optical input power of the light of the wavelength λj inducing the nonlinear effect, Leff the effective length,
                                          L            eff                    ⁡                      (                          λ              j                        )                          =                              1            -                          ⅇ                                                -                                      α                    ⁡                                          (                                              λ                        j                                            )                                                                      ⁢                L                                                          α            ⁡                          (                              λ                j                            )                                                          (        2        )            The index i represents the signal light and index j the pump light.
The attenuation α(λ) is wavelength dependent. The value of the constant b depends on the type of nonlinear effect (SPM or XPM), as well as on the state and evolution of the polarizations of signal (λi) and pump (λj) light. The nonlinearity coefficient is defined by (see non-patent document 4)
                                                                        γ                ⁡                                  (                                                            λ                      i                                        ,                                          λ                      j                                                        )                                            =                                                                    2                    ⁢                                                                                  ⁢                    π                                                        λ                    i                                                  ⁢                                                      ∫                                          ∫                                                                                                    n                            2                                                    ⁡                                                      (                                                          x                              ,                              y                                                        )                                                                          ⁢                                                                                                  ⁢                                                  I                          ⁡                                                      (                                                          x                              ,                              y                              ,                                                              λ                                i                                                                                      )                                                                          ⁢                                                                                                  ⁢                                                  I                          ⁡                                                      (                                                          x                              ,                              y                              ,                                                              λ                                j                                                                                      )                                                                          ⁢                                                                                                  ⁢                                                  ⅆ                          x                                                ⁢                                                  ⅆ                          y                                                                                                                          ∫                                          ∫                                                                        I                          ⁡                                                      (                                                          x                              ,                              y                              ,                                                              λ                                i                                                                                      )                                                                          ⁢                                                                                                  ⁢                                                  ⅆ                          x                                                ⁢                                                  ⅆ                          y                                                ⁢                                                                                                  ⁢                                                  ∫                                                      ∫                                                                                          I                                ⁡                                                                  (                                                                      x                                    ,                                    y                                    ,                                                                          λ                                      j                                                                                                        )                                                                                            ⁢                                                                                                                          ⁢                                                              ⅆ                                x                                                            ⁢                                                              ⅆ                                y                                                                                                                                                                                                                                                                                =                                                                    2                    ⁢                                                                                  ⁢                    π                                    λ                                ⁢                                                                                                    n                        _                                            2                                        ⁡                                          (                                                                        λ                          i                                                ,                                                  λ                          j                                                                    )                                                                                                  A                      eff                                        ⁡                                          (                                                                        λ                          i                                                ,                                                  λ                          j                                                                    )                                                                                                                              (        3        )            where n2(x,y) is the local nonlinear refractive index, I(x,y,λ) the optical intensity distribution for a certain wavelength λ,
                                                        n              _                        2                    ⁡                      (                                          λ                i                            ,                              λ                j                                      )                          =                              ∫                          ∫                                                                    n                    2                                    ⁡                                      (                                          x                      ,                      y                                        )                                                  ⁢                                                                  ⁢                                  I                  ⁡                                      (                                          x                      ,                      y                      ,                                              λ                        i                                                              )                                                  ⁢                                                                  ⁢                                  I                  ⁡                                      (                                          x                      ,                      y                      ,                                              λ                        j                                                              )                                                  ⁢                                                                  ⁢                                  ⅆ                  x                                ⁢                                  ⅆ                  y                                                                          ∫                          ∫                                                I                  ⁡                                      (                                          x                      ,                      y                      ,                                              λ                        i                                                              )                                                  ⁢                                                                  ⁢                                  I                  ⁡                                      (                                          x                      ,                      y                      ,                                              λ                        j                                                              )                                                  ⁢                                                                  ⁢                                  ⅆ                  x                                ⁢                                  ⅆ                  y                                                                                        (        4        )            is the average nonlinear refractive index, and
                                          A            eff                    ⁡                      (                                          λ                i                            ,                              λ                j                                      )                          =                              ∫                          ∫                                                I                  ⁡                                      (                                          x                      ,                      y                      ,                                              λ                        i                                                              )                                                  ⁢                                                                  ⁢                                  ⅆ                  x                                ⁢                                  ⅆ                  y                                ⁢                                                                  ⁢                                  ∫                                      ∫                                                                  I                        ⁡                                                  (                                                      x                            ,                            y                            ,                                                          λ                              j                                                                                )                                                                    ⁢                                                                                          ⁢                                              ⅆ                        x                                            ⁢                                              ⅆ                        y                                                                                                                                      ∫                          ∫                                                I                  ⁡                                      (                                          x                      ,                      y                      ,                                              λ                        i                                                              )                                                  ⁢                                                                  ⁢                                  I                  ⁡                                      (                                          x                      ,                      y                      ,                                              λ                        j                                                              )                                                  ⁢                                                                  ⁢                                  ⅆ                  x                                ⁢                                  ⅆ                  y                                                                                        (        5        )            is the so called effective area.
In the case of FWM the overlap integrals in (3) are taken over spatial field distributions of the four wavelengths involved (see non-patent document 5).
[Non-Patent Document 1]
    Photonics Technology Letter vol. 15, no. 3 Mar. 2003 p437[Non-Patent Document 2]    Photonics Technology letters, vol. 15, no. 3, March 2003 p440[Non-Patent Document 3]    “Nonlinear fiber optics”, Academic Press, 2nd edition, (1995)[Non-Patent Document 4]    T. Kato, Y. Suetsugu, and M. Nishimura, “Estimation of nonlinear refractive index in various silica-based glasses for optical fibers”, Opt. Lett., 20(22): 2279-2281 (1995)[Non-Patent Document 5]    G. Agrawal, “Nonlinear fiber optics”, Academic Press, 2nd edition, (1995) ch. 10, p. 408
FIG. 1 shows a cross sectional view of a square-lattice hole structured photonic crystal fiber.
The wavelength dependence of the nonlinear phase shift is exemplarily studied for XPM in a PCF with a square-lattice hole structure as depicted in FIG. 1. The core is formed by displacing the four center holes outwards by δ=0.1Λ. The ratio between the hole-diameter d and the lattice period Λ is set to d/Λ=0.9. The silica glass matrix is assumed to be F-doped with 1 wt-%, which increases the nonlinear refractive index by a factor of 1.4 compared to pure silica [T. Kato, Y. Suetsugu, and M. Nishimura, “Estimation of nonlinear refractive index in various silica-based glasses for optical fibers”, Opt. Lett. 20(22): 2279-2281].
In the structure of FIG. 1, light is confined in area A but penetration of light outside the area A occurs. In order to calculate the nonlinear phase shift defined by (1) and (3) a finite element method is employed for determining the optical intensity distribution in the PCF. In this example the wavelength of the signal and the pump light are set less than 2 nm apart. Thus, the intensity distributions of the two wavelengths can be assumed to be identical.
FIG. 2 plots the relative nonlinear phase shift over a range of 40 nm for various lattice periods, which correspond to different core sizes and thus different values of the nonlinearity coefficient. The results reveal that with decreasing core size the wavelength dependence of the relative nonlinear phase shift φ(λ)/φ(1550 nm) increases. For a γ of 72 W−1km−1 the relative change amounts to 6.5% over a wavelength range of 40 nm.
In the above example, it was assumed that pump and signal wavelength propagate with nearly the same group velocity, i.e., the walk-off between the two wavelengths is negligible. Moreover, it was assumed that the coupling efficiency into the PCF and the transmission loss are wavelength independent.
As described above, in the PCFs, there is wavelength dependence of the nonlinear phase shift. As the nonlinear phase shift corresponds to the efficiency of PCFs as nonlinear device, the difference in nonlinear phase shifts for different wavelengths induces the difference in the efficiency of nonlinear device for different wavelengths, which is made from PCFs. When considering using the PCFs for nonlinear devices in the wavelength division multiplexing (WDM) transmission system, this causes the problem.